Departure time choice equilibrium problem with
partial implementation of congestion pricing
Tokyo Institute of Technology
Postdoctoral researcher
Katsuya Sakai
1
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
2
Problems of congestion
• wasting a lot of resources
– Time
– Energy
– Mental health
–…
3
How congestion occurs?
• caused by a concentration of demand
– At a location
origin
destination
– During short period
Demand rate
Time of day
4
How to eliminate congestion?
• distributing the concentrated demand
– At a location
origin
destination
– During short period
Demand rate
Time of day
5
Congestion management
• Information provision
– Congestion length
– Travel time
• Physical control (quantitative control)
– Ramp metering
– Advance booking
• Economic approach (pricing control)
– Congestion charging
– Tradable bottleneck permits
6
Congestion management
Approach
Information
provision
Quantitative
control
Pricing
control
Advantages
Limitations
Travel time can be reduced
Total travel time cannot
be minimized
Total travel time can be
minimized
Social cost cannot be
minimized
Social cost can be
minimized
Equity problem; poor
people may suffer a loss
• Focusing on pricing control (economic approach)
- Tradable bottleneck permits (TBP) scheme
7
Tradable bottleneck permits scheme
(Akamatsu et al. 2006)
1. Road administrators issue a right that allows a
permit holder to pass through the bottleneck at
a pre-specified time period (“bottleneck
permits”).
2. A new auction market is established for
bottleneck permits differentiated by a prespecified time.
– As a result of auction, toll is determined by each
commuter’s willingness to pay (value-pricing)
– Under the toll, social cost is minimized
8
Equity problem of congestion charging
• The Welfare Effects of Congestion Tolls with
Heterogeneous Commuters (Arnott et al.,
1994)
– The poor suffer a loss
– The rich get a benefit
9
Suggestion
• Applying congestion pricing only a portion of
road (lanes).
– Drivers can choose
paying or not paying
Free
Toll
• How much portion should be charging lanes?
• Don’t the poor suffer a loss?
10
How to evaluate pricing scheme?
• Comparing equilibrium states before/after
pricing applied, we can know the effect of
congestion pricing.
Equilibrium
AFTER pricing
- Commuter’s travel cost
- Social cost
Equilibrium
BEFORE pricing
- Commuter’s travel cost
- Social cost
Need to calculate equilibrium
11
Objective
• To formulate the departure time choice
equilibrium problem and to solve it when the
TBP scheme is partially applied.
• To examine the welfare effect of partial
congestion pricing with heterogeneous
commuters.
12
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
13
What is equilibrium?
• The most realizable state.
• At equilibrium, nobody can improve their
travel cost by unilaterally changing their
behavior.
Equilibrium
14
What is equilibrium?
• Route choice equilibrium (Wardrop, 1952)
– The journey times on all the routes actually used
are equal, and less than those which would be
experienced by a single vehicle on any unused
route.
• Departure time choice equilibrium (Vickrey,
1969)
– Times => costs
– Routes => departure time
15
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
16
Route choice equilibrium
(Wardrop, 1952)
• Both routes are used
– Travel time of two routes must be same
• Either of routes is used
– Travel time of used road must be smaller or at
most equal to that of the unused road.
17
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
18
Departure time choice problem
(Vickrey, 1969)
• Focusing on morning peak hour (commute) problem
• Employing bottleneck model, we simply describe
traffic congestion one-to-one network with a single
bottleneck
Demand: N
Destination
Origin
(Demand rate)
>=
<
(Capacity)
19
Departure time choice equilibrium
(Vickrey,
1969)
Travel cost /
Demand rate
Travel cost
Demand rate
Time of day
Time
without demand
Time with demand
Time
without demand
20
Departure time choice equilibrium
(Vickrey,
1969)
Travel cost /
Demand rate
What is travel cost?
Travel cost
How to determine?
Demand rate
Time of day
Time
without demand
Time with demand
Time
without demand
21
Departure time choice equilibrium
(Vickrey, 1969)
• Defining travel cost with tradeoff between waiting
and schedule delays.
Waiting cost
Schedule cost
Toll cost
Value of time
In working hour
home
office
out of working hour
Driving
γ
β
α
TC: travel cost
w: waiting delay
α: marginal cost of waiting delay
β: marginal cost of early arrival
γ: marginal cost of late arrival
t0: desired arrival time
t: actual arrival time
22
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
23
Equilibrium conditions without toll
• Departure (arrival) time choice equilibrium condition;
∙
if
0
∀
∙
if
0
• Demand-supply equilibrium condition;
if
0
∀
if
0
• Flow conservation
TC: travel cost
• Non-negativity constraint
0∀
: equilibrium travel cost
q: demand rate
μ : bottleneck capacity
w: waiting delay
: total demand
24
How to solve equilibrium problem?
• Solving the equivalent optimization problem (Iryo and
Yoshii, 2007)
• Minimizing total schedule cost (waiting-time base)
– KKT conditions are equal to the original problem
*It is also possible to analytically solve the equilibrium problem.
25
Equilibrium solution without toll
Cumulative
# of Vehicles
N
D(t)
A(t)
Cost
t0
Time of day
TC
!∙" #
$ #
t0
Time of day
26
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
27
Equilibrium conditions with toll (TBP)
• Departure (arrival) time choice equilibrium condition;
%
if
0
∀
%
if
0
• Demand-supply equilibrium condition;
if%
0
∀
if%
0
• Flow conservation
TC: travel cost
• Non-negativity constraint
0∀
: equilibrium travel cost
p : price of bottleneck permits
q: demand rate
μ : bottleneck capacity
p: price of TBP
N : total demand
28
Equivalent optimization problem
– Minimizing total schedule cost (monetary base)
*It is also possible to analytically solve the equilibrium problem.
29
Equilibrium solution with TBP
Cumulative
# of Vehicles
N
D(t)
A(t)
Cost
ts
t0
TC
$ #
ts
tf
Time of day
tf
Time of day
' #
t0
30
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
31
Departure time choice equilibrium problem with partial
implementation of tradable bottleneck permits
• Problem including departure time choice and
lane (route) choice problem
• Commuter’ heterogeneity
– Value of time (waiting delay cost)
– Flexibility of job (schedule delay cost)
route 1
µ(1)
(without TBP)
O
D
µ
O
D
route 2
(with TBP)
µ(2)
32
Equilibrium conditions
•
•
•
•
•
Departure (arrival) time and route choice equilibrium condition;
()*
if ()*
0
∀+,
()*
()*
if
0
Demand-supply equilibrium condition in route 1;
(-*
if
0
∀
if
0
Demand-supply equilibrium condition in route 2;
.
.
if%
0
∀
.
.
if%
0
Flow conservation
Non-negativity constraint
()*
)
)
0∀+,
33
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
34
How to solve?
• No equivalent optimization problem
• Dividing the problem into 3 parts
Route assignment problem
Departure time choice
Equilibrium in route 1:
LP optimization problem
route 1
Nk
µ(1)
Departure time choice
Equilibrium in route 2:
LP optimization problem
Nk(1)
(without TBP)
route 1
µ(1)
(without TBP)
O
O
D
D
route 2
(with TBP)
O
µ(2)
Nk(2)
route 2
(with TBP)
D
µ(2)
35
Solution algorithm
start
input parameters and variables
Nk, αk, βk, γk, µ(1), µ(2)
route assignment
xk
Solving departure
time choice
equilibrium in each
route
(1-xk)Nk, αk, βk, γk, µ(1)
xkNk, βk, γk, µ(2)
departure time choice
equilibrium in route 1
departure time choice
equilibrium in route 2
q(1)k(t), w(t), TC
ρk k(1)
Route choice
equilibrium condition
TC (1) < TC ( 2 ) if x = 0
k
k
 k
 (1)
( 2)
TCk = TCk if 0 < xk < 1
 (1)
( 2)
TCk > TCk if xk = 1

q(2)k(t), p(t), TC
πk k( 2)
discriminant of
route choice equilibrium
No
Yes
∀k
output variables
q(2)k(t), w(t), p(t), TC
Ukk
q(1)k(t),
end
36
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
37
Input
• Bottleneck capacity: μ=100 veh/min
• Capacity without toll: μ(1)=40 veh/min
• Capacity with toll: μ(2)=60 veh/min
Model inputs
Nk (veh)
k=1 1000
k=2 2000
k=3 3000
t0
8:50
9:10
9:00
αk (USD/min)
0.25
1.00
0.50
βk (USD/min)
0.20
0.10
0.15
γk (USD/min)
0.30
0.20
0.40
38
Convergence to equilibrium
Rate of choosing route 2
(Buying bottleneck permits)
1
group 1
group 2
group 3
0.5
0
0
10
20
30
Iteration
39
Convergence to equilibrium
Route 1 (without toll)
Route 2 (with toll)
10
10
group 1
group 2
group 3
5
0
Travel cost in route 2 (USD)
Travel cost in route 1 (USD)
15
group 1
group 2
group 3
5
0
0
10
20
30
0
10
20
30
Iteration
Iteration
40
Equilibrium state
Waiting delay
Price of TBP
Price of bottleneck permit (USD)
Waiting delay (min)
15
10
5
0
8:20
8:30
8:40
8:50
9:00
Time of day
9:10
9:20
9:30
6
4
2
0
8:20
8:30
8:40
8:50
9:00
Time of day
9:10
9:20
9:30
41
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
42
Welfare effect analysis
• Examining changes in commuter’s route
choice and travel cost.
Full pricing (ρ=100%)
All of commuters
need to buy TBP
Partial pricing
(ρ=90%)
Partial pricing
(ρ=10%)
No pricing (ρ=0%)
No commuters
need to buy TBP
43
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
44
Welfare effect analysis
Input
• Commuter’s heterogeneity in value of time and flexibility of job
(Job flexibility)
(value of time)
Poor
Rich
k=1
k=2
k=3
k=4
qk (veh)
1000
1000
1000
1000
α=0.20
α=1.00
t0
9:00
9:00
9:00
9:00
Flexible Inflexible
δ=0.20
δ=0.80
[Group 1] [Group 2]
[Group 3] [Group 4]
αk (USD/min)
0.20
0.20
1.00
1.00
δ: flexibility of job (δ β/α)
β/γ assumed constant
δ
0.20
0.80
0.20
0.80
βk (USD/min)
0.04
0.16
0.20
0.80
γk (USD/min)
0.08
0.32
0.40
1.60
45
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
46
Change in route choice
Rate of choosing route 2
(Buying bottleneck permits)
1
group 1
group 2
group 3
group 4
0.5
0
0
Flexible
Inflexible
Poor
group 1
group 2
Rich
group 3
group 4
0.2
0.4
0.6
0.8
TBP implementation rate
1
47
Change in travel cost
Equilibrium travel cost (USD)
15
10
group 1
group 2
group 3
group 4
5
0
0
Flexible
Inflexible
Poor
group 1
group 2
Rich
group 3
group 4
0.2
0.4
0.6
0.8
TBP implementation rate
1
48
Contents
1.
2.
Introduction
Method/Tool (Previous studies)
1.
2.
3.
Equilibrium
Route choice equilibrium
Departure time choice equilibrium
1.
2.
3.
Departure time choice equilibrium problem with partial implementation
of tradable bottleneck permits (My study)
1.
2.
4.
Solution procedure
Numerical example
Welfare effect analysis (My study)
1.
2.
5.
Without toll
With toll
Input
Result
Conclusion
49
Conclusions
• Equilibrium problem under partial TBP
implementation
• Welfare effect of partial congestion pricing
scheme
• Possibility of congestion pricing scheme
harming nobody even if the toll revenues are
not refunded
• Future works
– To prove uniqueness of equilibrium point
50
References
• Arnott, R. J.; de Palma, A. & Lindsey, R. The Welfare Effects
of Congestion Tolls with Heterogeneous Commuters Journal
of Transport Economics and Policy, University of Bath and
The London School of Economics and Political Science, 1994,
28, 139-161.
• Iryo, T. and Yoshii, T., Equivalent optimization problem for
finding equilibrium in the bottleneck model with time
choices, in Mathematics in Transport, B.G. Heydecker (Ed.),
2007, Elsevier: Oxford. 231-244.
• Vickrey, W. S. Congestion Theory and Transport Investment
The American Economic Review, 1969, 59, 251-260
• Wardrop, J. G. Some Theoretical Aspects of Road Traffic
Research Proceedings of the Institution of Civil Engineers,
1952, 1, 325-362.
51